For a variable line $\frac{x}{a} + \frac{y}{b} = 1$ where $a + b = 10$,find the equation of the locus of the midpoint of the portion of the line intercepted between the coordinate axes.

Find the fixed point through which the line $x(a + 2b) + y(a + 3b) = a + b$ always passes for all values of $a$ and $b$.

$A$ variable straight line passes through the point of intersection of the lines $x + 2y = 1$ and $2x - y = 1$ and meets the coordinate axes at $A$ and $B$. The locus of the midpoint of $AB$ is:

Two particles $A$ and $B$ move from rest along a straight line with constant accelerations $f$ and $h,$ respectively. If $A$ takes $m$ seconds more than $B$ and describes $n$ units more than that of $B$ to acquire the same speed, then

If the sum of distances from a point $P$ to two mutually perpendicular straight lines is $1$ unit, then the locus of $P$ is

If a variable straight line passing through the point of intersection of the lines $x-2y+3=0$ and $2x-y-1=0$ intersects the $X, Y$-axes at $A$ and $B$ respectively,then the equation of the locus of a point which divides the segment $AB$ in the ratio $-2:3$ is